Environmental feedback sign shapes cooperation and collapse in eco-evolutionary networks

Cooperation in a system with a shared renewable resource is shaped by a feedback loop: individual actions alter the resource, which in turn changes the payoff to cooperation. We study this loop in a spatial evolutionary game with a signed feedback strength λ. For an isolated interior fixed point on the differentiable, unclipped branch of the mutation-free mean-field model, negative feedback (λ < 0) makes the fixed point locally attracting, whereas positive feedback (λ > 0) makes it a saddle. This local classification provides a reference for the spatial dynamics. At the sampled negative-feedback operating points, demographic noise produces quasi-cycles: their amplitude scales as N-1/2, the rescaled spectra collapse with system size, and resource fluctuations lag cooperation by approximately 1 rad. Under positive feedback, runs initialized in the prosperous and desert states remain strongly separated over the observation window, and recovery slows near the loss of prosperity. Resource diffusion can reverse the motion of a depleted patch from retreat to invasion, with the two regimes separated by a zero-velocity boundary. In multiplex systems, weak mass-conserving exchange can rescue a damaged layer, whereas stronger exchange can spread the deficit and promote joint collapse. The same qualitative contrast persists on the tested square lattices, small-world networks, and scale-free networks under cumulative payoffs. The results show how the sign of a strategy-resource feedback organizes fluctuations, spatial invasion, and interdependent failure.